Remarks on Hilbert Identities, Isometric Embeddings, and Invariant Cubature

Algebra i analiz St. Petersburg Math. J. Tom 25 (2013), 4 Vol. 25 (2014), No. 4, Pages 615–646 S 1061-0022(2014)01310-6 Article electronically published on June 5, 2014

REMARKS ON HILBERT IDENTITIES, ISOMETRIC EMBEDDINGS, AND INVARIANT CUBATURE

H. NOZAKI AND M. SAWA

Abstract. In 2004, Victoir developed a method to construct cubature formulas with various combinatorial objects. Motivated by this, the authors generalize Victoir’s method with yet another combinatorial object, called the regular t-wise balanced design. Many cubature formulas of small indices with few points are provided, which are used to update Shatalov’s table (2001) of isometric embeddings in small-dimensional Banach spaces, as well as to improve some classical Hilbert identities. A famous theorem of Bajnok (2007) on Euclidean designs invariant under the Weyl group of Lie type B is extended to all ﬁnite irreducible reﬂection groups. A short proof of the Bajnok theorem is presented in terms of Hilbert identities.

§1. Introduction Let p be a positive integer such that p = ∞.Them-dimensional Euclidean space Rm m is a Banach space lp endowed with the norm m 1/p p xp = |xi| . i=1

m n Given two spaces lp and lq , a classical problem in Banach space theory asks when there R m → n is an -linear map F : lp lq such that

F (x)q = xp

∈ m m n for every x lp . Suchamapiscalledanisometric embedding from lp to lq . To exclude trivial cases, we assume that n ≥ m ≥ 2andp = q.Itisknown[22,Theorem1.1]thatif ∞ m n p, q = and an isometric embedding from lp to lq exists, then p =2andq is an even integer. Throughout this paper we only consider the case where p =2andq is even, and ﬁx the notations p, q, m, n. m 2 q/2 Isometric embeddings are closely related to representations of i=1 xi as a sum of qth powers of linear forms with positive real coeﬃcients. Such representations originally stem from a work of Hilbert on Waring’s problem [16], and are therefore called Hilbert identities [25]. Hilbert solved Waring’s problem, showing on the way that there m → n exist isometric embeddings l2 lq with n depending on m and q. Several alternative proofs of Hilbert’s theorem are known; for example, see [6, 7] and the references therein.

2010 Mathematics Subject Classiﬁcation. Primary 65D32, 11E76; Secondary 52A21. Key words and phrases. Cubature formula, Hilbert identity, isometric embedding, Victoir method. The second author was supported in part by Grant-in-Aid for Young Scientists (B) 22740062 and Grant-in-Aid for Challenging Exploratory Research 23654031 by the Japan Society for the Promotion of Science.

c 2014 American Mathematical Society 615

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But most of them, including the original proof by Hilbert, involve nonconstructive arguments in analysis, and do not give any explicit constructions of embeddings.1 Thus, publications with explicit embeddings continued to appear. Isometric embeddings are also related to a certain object in numerical analysis. Let Ω be a subset of Rm on which a normalized measure μ is deﬁned. A ﬁnite subset X of Ω with a positive weight w is called a cubature formula of index q if (1.1) f(x)μ (dx)= w(x)f(x) Ω x∈X

for every f ∈ Homq(Ω), where Homq(Ω) is the space of all homogeneous polynomials of degree q restricted to Ω. Lyubich and Vaserstein [22] and Reznick [27] proved the m → n equivalence between an embedding l2 lq and an n-point cubature formula of index q for the surface measure ρ on the (m − 1)-dimensional unit sphere Sm−1. Many papers are devoted to the construction of spherical cubature formulas. There are two classical approaches. One employs orbits of finite subgroups of the orthogonal group O(m) acting on Sm−1 [33], and the other takes a “product” of several lower-dimensional cubatures [34]. Cubature formulas that are studied in the context of numerical analysis and related areas, are often of degree type. Victoir [35] developed a novel technique to construct degree-type cubature for integrals with special symmetry. His idea is as follows. Given a cubature formula invariant under the Weyl group of Lie type B, one eliminates some specified points of the formula by using combinatorial objects such as t-designs and orthogonal arrays. With this method, Victoir found many cubature formulas of small degrees with few points in general dimensional spaces. In this paper, we have several important aims. First, we generalize the Victoir method to a special class of block designs, called regular t-wise balanced designs. The concept of aregulart-wise balanced design has been substantiated by applications in statistics [8, 10, 18], however, it seems that there is insufficient evidence to support it from other mathematical aspects. To find a new meaning of this concept, as well as to let researchers in many areas of mathematics know it, are among our important aims in this paper. On the other hand, Bajnok [1, Theorem 3] proved that Euclidean designs, a generalization of the spherical cubatures, that are invariant under the Weyl group of Lie type B have degree at most 7. We further discuss the Bajnok theorem both from a combinatorial and analytic point of view. This paper is organized as follows. In §2 we review some basic facts and notions, and explain the Victoir method in detail. In §3 we generalize the Victoir method to regular t-wise balanced designs. In §4, we give general-dimensional index-four and -six cubature formulas, together with some additional examples of index-six cubature formulas that m → n § improve Shatalov’s table [32, Theorem 4.7.20] of isometric embeddings l2 l6 .In 5, we generalize the Bajnok theorem for all finite irreducible reflection groups, and thereby classify the spherical cubature formulas with a certain geometric meaning. In §6, some of the cubatures constructed in §4and§5 are translated into Hilbert identities, in order to improve classical identities as those by Schur [6] and Reznick [27]. An extremely short proof of the Bajnok theorem is given in terms of Hilbert identities.

§2. Preliminaries 2.1. Isometric embeddings and Hilbert identities. Lyubich and Vaserstein [22] and Reznick [27] observed a close relationship between Hilbert identities, isometric embeddings, and spherical cubature formulas.

1Bruce Reznick kindly told us that Stridsberg’s proof (1912) is constructive, provided we know how to compute the roots of Hermite’s polynomials.

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Theorem 2.1. The following statements are equivalent. (i) There exists a cubature formula of index q on Sm−1 with n points. m → n (ii) There exists an isometric embedding l2 lq . m m (iii) There exist n vectors r1,...,rn ∈ R such that, for any x ∈ R , n q q x, x 2 = x, ri . i=1 We explain Theorem 2.1 in detail for further arguments in the following sections. m−1 Assume that points x1,...,xn ∈ S and weights w1,...,wn form a cubature of index m−1 q m q on S .Letx, y ∈ Homq(R ), where ·, · denotes the usual inner product. Then n q q q wix, xi = x, y ρ (dy)=x, x 2 cq, m−1 i=1 S where q cq = y1ρ (dy),y=(y1,...,ym). Sm−1 This is, equivalently, n q q x, x 2 = x, ri , i=1 q where ri = wi/cqxi. This polynomial identity is further transformed as follows:

1 n q 1 q x, x 2 = x, ri , i=1 which implies that the mapping

x → (x, r1 ,...,x, rn ) m → n is an isometric embedding l2 lq . By the early fundamental works of Hilbert [16], there is a positive integer N(m, q) ≥ m → n such that for any n N(m, q) an isometric embedding l2 lq exists.Itisknown (cf. [27]) that m + q − 1 m + q − 1 (2.1) 2 ≤ N(m, q) ≤ . m − 1 m − 1 m The lower- and the upper-bound part of (2.1) mean the dimension of Homq/2(R )and m Homq(R ), respectively. 2.2. Cubature formulas. Let Ω ⊂ Rm, and let μ be a normalized measure on Ω such that Ω and μ are both invariant under the group O(m). We assume that polynomials are integrable up to suﬃciently large degrees and put I[f]= f(x)μ (dx). Ω Let X be a ﬁnite set in Rm with a positive weight w. The pair (X, w) is called a cubature formula of degree q for I if I[f]= w(x)f(x) x∈X

for every f ∈Pq(Ω), where Pq(Ω) denotes the space of all polynomials of degree at most q restricted to Ω. In particular, a spherical cubature is called a spherical design if w is a constant weight.

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A subset X of Rm is said to be antipodal if it is partitioned into X and −X ,namely, X = X ∪ (−X )andX ∩ (−X )=∅. A cubature formula (X, w)iscentrally symmetric if X is antipodal and w(x)=w(−x) for any x ∈ X. In the following, we mention the relationship among degree-type and index-type spherical cubature formulas. Proposition 2.2 (see [22, Proposition 4.3]). .LetX be an antipodal ﬁnite subset of Sm−1.ThenX is a centrally symmetric cubature formula on Sm−1 of degree q +1 with 2n points if and only if X is a cubature formula on Sm−1 of index q with n points. We are interested in the following type of integrals:

(2.2) f(x)W (x2)dx, Rm where W is a density function on Rm. Such integrals are often considered in the context of analysis; for example, see [38].

Proposition 2.3. If points x1,...,xn and weights w1,...,wn form a cubature formula x q w i 2 i of index q for (2.2), then the points xi/ xi 2 and the weights ∞ q+m−1 form a cu- 0 r W (r)dr m−1 bature formula of index q on S . Conversely, if x1,...,xn and w1,...,w n form a cuba- m−1 ∞ q+m−1 ture formula of index q on S , then the points xi and the weights wi 0 r W (r)dr form a cubature formula of index q for (2.2). m Proof. The result follows by observing that for any f ∈ Homq(R )wehave ∞ m−1 f(x)W (x2)dx = f(rx)ρ(dx) r W (r)dr Rm 0 Sm−1 ∞ = rq+m−1W (r)dr f(x)ρ (dx). 0 Sm−1 Remark 2.4. By Proposition 2.3, in order to construct spherical cubature formulas we may find those for any integral of the form (2.2). For example, one may think of Gaussian integrals. Such cubature formulas are of particular interest in probability theory [21] and algebraic combinatorics [2]. Moreover, the m-dimensional Gaussian integral can be represented simply as the m-fold product of one-dimensional Gaussian integrals, which is convenient for explaining Victoir’s method. The following proposition is often used in §3and§4. Proposition 2.5. Let X be an antipodal finite subset of Rm.Letw, w be weight functions on X and X, respectively, such that w(x)=w(−x)=2w (x) for any x ∈ X .Then(X, w ) is a cubature formula of index q for (2.2) if and only if (X, w) is a centrally symmetric cubature formula of index q for (2.2). m 2.3. The Sobolev theorem. Let G be a finite subgroup of O(m), and let f ∈Pt(R ). We define the action of σ ∈ G on f as follows: −1 (σf)(x)=f(xσ ),x∈ Rm. A polynomial f is said to be G-invariant if σf = f for every σ ∈ G.Wedenotetheset m m m G m G of G-invariant polynomials in Pt(R )andHarmt(R )byPt(R ) and Harmt(R ) , m m respectively, where Harmt(R ) is the subspace of Pt(R ) of harmonic homogeneous polynomials of degree t. A cubature formula is said to be G-invariant if the domain and measure of the integral G G are invariant under G, the points form a union of G-orbits z1 ,...,ze ,andw(x)=w(x ) ∈ G G G for any x, x zi ; the orbits z1 ,...,ze and weights w1,...,we are said to generate the formula.

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Theorem 2.6 (see [33]). With the above setup, a G-invariant cubature formula is of m G degree t if and only if it is exact for every polynomial f ∈Pt(R ) . Theorem 2.6 is known as the Sobolev theorem, which is at the core of the Victoir method, as will be seen in the next subsection. The concept of Euclidean designs was introduced by Neumaier and Seidel [23] as a generalization of spherical cubature formulas. Let X be a ﬁnite set in Rm, and let { | ∈ } { } m−1 x 2 x X = r1,...,rp .LetSi be the sphere of radius ri centered at the ∪p m−1 ∩ m−1 origin, let S = i=1Si , and let Xi = X Si .Toeach Si, the surface measure m−1 1 | | − − ρi is assigned. Let Si = Sm 1 ρi (dx), where | m−1| Sm 1 f(x)ρi (dx)=f(0) if i Si i m−1 { } Si = 0 . Deﬁnition 2.7 (see [23]). With the above setup, X is a Euclidean t-design of Rm if p w(x) x∈Xi (2.3) m−1 f(x)ρi (dx)= w(x)f(x) |S | m−1 i=1 i Si x∈X

for every polynomial f ∈Pt(S). As is readily seen by the definition, the Euclidean designs can be viewed as cubature formulas on multiple concentric spheres. The following is a variation of the Sobolev theorem for Euclidean designs, which generalizes the well-known theorem of Neumaier and Seidel [23]. ∪M G Theorem 2.8 (see [24]). Let G be a subgroup of O(m).LetX = k=1rkxk ,where m−1 xk ∈ S and rk > 0. Then the following statements are equivalent: m (i) Xis a G-invariant Euclidean t-design of R ; 2j ∈ Rm G ≤ ≤ ≤ ≤t−l (ii) x∈X w(x) x ϕ(x)=0for any ϕ Harml( ) , 1 l t, 0 j 2 . Hereafter, let G be an irreducible reflection group in Rm. Such groups are classified completely [4]. Let integers 1 = d1 ≤ d2 ≤ ··· ≤ dm be the exponents of G (see [4, Chapter V, §6]). Theorem 2.9 (see [9]). Let G be a finite irreducible reflection group. Let m G qi = dim(Harmi(R ) ). Then ∞ m i 1 qiλ = . 1 − λ1+di i=0 i=2 m G m−1 In particular, for any x ∈ R , the orbit x is a spherical d2-design in S .

Let α1,...,αm be the fundamental roots of a reﬂection group G.Thecorner vectors v1,...,vm are deﬁned by vi ⊥ αj if and only if i = j. We may assume that vk2 =1. We consider the set X G (G, J)= rkvk , k∈J

where J ⊂{1, 2,...,m} and rk > 0. Let R denote the set of all rk. Theorem 2.10 (Bajnok [1, Theorem 3]). Let m ≥ 2 be an integer. Then there is no m choice of R, J,andw for which (X (Bm,J),w) is a Euclidean 8-design of R .

Similar results are known for the groups Am−1,Dm [24]. In §5, we generalize these results, and determine the maximal degree of invariant Euclidean designs for all irreducible reﬂection groups.

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2.4. The Victoir method.

2.4.1. Combinatorial tools. Let K be a set of positive integers k1,...,k.Apairof v elements of V and subsets B of V of cardinalities from K is called a t-wise balanced design, denoted by t-(v, K, λ), if every t elements of V occur exactly λ times in B.The elements of V and B are called points and blocks. In particular, if K is a singleton, say K = {k},at-wise balanced design is called a t-design, and is denoted by t-(v, k, λ). In this paper we only consider designs without repeated blocks. It is well known (cf. [19]) that for 0 ≤ t ≤ t and a subset T ⊂ V of t elements, the number of blocks of a t-(v, k, λ) design containing T is given by v−t t−t (v − t )(v − t − 1) ...(v − t +1) (2.4) λ = , k−t (k − t)(k − t − 1) ...(k − t +1) t−t not depending on the choice of T .Foreach0≤ t ≤ t,at-design is also a t-design. In general, t-wise balanced designs do not necessarily have this property; see §3forthe details. Let (V,B)beat-wise balanced design with v points and b blocks. The incidence matrix M of the design (V,B) is a zero-one matrix of size v × b which has a row for each point and a column for each block, and for x ∈ V and B ∈B,the(x, B)-entry takes 1 if and only if x ∈ B. Given real numbers α, β,letvl(α, β)beav-dimensional vector such that the first l coordinates are α and the remaining v − l coordinates are β.For Bv example, vl(α, 0) means the vertices of a generalized√ hyperoctahedron that is inscribed in the (v − 1)-dimensional sphere of radius lα2 [1]. With the matrix M, we associate a generalized incidence matrix with parameters α, β by defining Iα,β = βJv,b +(α − β)M, where α = β and Jv,b is the all-one matrix of size v × b. An (N × l)-matrix with entries ±1 is called an orthogonal array with strength t,con- straints l,andindexλ if in every t columns, each of the 2t ordered combinations of elements ±1 appears in exactly λ rows.WedenotethismatrixbyOA(N,l,2,t). We do not put λ in the notation, because λ = N/2t by the definition. When l ≤ t,we allow trivial OA,namely,the(2l × l)-matrix such that every 2l ordered combinations of elements ±1 appears in exactly one row.

2.4.2. Victoir’s method. The group Bm contains two special subgroups: the subgroup L of all transpositions of coordinates in Rm and the subgroup L´ of all sign changes, which is isomorphic to the elementary Abelian 2-group (Z/2Z)m. It turns out that |yL´ | =2|wt(y)|,wherewt(y) is the number of nonzero coordinates of a vector y. We denote by I the Gaussian integral 1 x2 I[f]= f((x2,...,x2 )) exp − 2 dx ... dx . m/2 1 m 1 m (2π) Rm 2 I´ Rm This is equivalent to the integral on the ﬁrst orthant + : − 1 x m 1/2 I´[f]= f((x ,...,x )) exp − 1 x dx ... dx . m/2 1 m i 1 m (2π) Rm 2 + i=1 Let 2 2 2 x =(x1,...,xm) m for x =(x1,...,xm) ∈ R , and let √ √ √ x = x1,..., xm ∈ Rm for x =(x1,...,xm) + .

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L´ L´ ´ Proposition 2.11 (cf. [35, 39]). If z1 , ..., ze and w1, ..., we generate an L-invariant I 2 2 wt(z1) wt(ze) cubature formula of degree q for ,thenz1 , ..., ze and w12 , ..., we2 form a cubature formula of degree q/2 for I´. Conversely, if z1, ..., ze and w1, ..., we √ ´ √ ´ L L wt(z1) form a cubature formula of degree q/2 for I´,then z1 , ..., ze and w1/2 , ..., wt(ze) we/2 generate a cubature formula of degree q for I. The following theorem is due to Victoir [35, Subsection 4.4]. Theorem 2.12. (i) Assume that there exists a cubature formula of degree q/2 for I´ of the form w M w I´[f]= f(x)+ i f(x), m |xL| k ∈ L i=1 i ∈ L x vk(α,β) x xi and a q/2-design with m points and b blocks of size k.LetX be the columns of a generalized incidence matrix with parameters α, β.Then w M w I´[f]= f(x)+ i f(x) b |xL| x∈X i=1 i ∈ L x xi is a cubature formula of degree q/2. (ii) Assume that there exists an L´-invariant cubature formula of degree q for I of the form M λ I[f]= i f(x), 2wt(xi) i=1 ∈ L´ x xi and OA(|Xi|, wt(xi), 2,q) with rows Xi for i =1,...,M.Then M λ I[f]= i f(x) |Xi| i=1 x∈Xi is a cubature formula of degree q. The Victoir method was originally written in a more general setting, namely, the integrals considered there were not restricted to Gaussian integrals. In this paper, however, we take only Gaussian integrals because Victoir’s ideas can be fully understood with Gaussian integrals.

§3. Generalizing the Victoir method In this section we generalize the Victoir method with a strengthening of the concept of t-wise balanced designs. We use the notations Bm,L,L´, I, I´, vi(·, ·), wt(·)thatare deﬁned in Subsection 2.4. A t-wise balanced design (V,B)issaidtoberegular if for each 0 ≤ t ≤ t and each t-subset T of V , the number of blocks containing T does not depend on the choice of T ; see [8]. As was noted in Subsection 2.4, any t-design possesses this property, but t-wise balanced designs may fail to do so. When t =2,thisconceptisequivalenttothat of equireplicate 2-wise balanced designs [10]. Let B be the set of blocks of a regular t-(v, K, λ) design, where K = {k1,...,kf }.Let m Bi = {B ∈B||B| = ki}.Letyi ∈ R with wt(yi)=ki,andyK = {y1,...,yf }.We deﬁne the following discrete measure: f |B | δ := i δ . yK ,L |B| m x i=1 ki ∈ L x yi

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Proposition 3.1. Assume that there exists a regular t-(m, {ki | 1 ≤ i ≤ f},λ) design (V,B).LetX be the columns of a generalized incidence matrix with parameters α, β with α = β.Lety1,...,yf ∈ X be such that wt(yi)=ki.Then 1 f(x)δyK ,L (dx)= f(x) f L |B| i=1 yi x∈X ∈P f L for every f t( i=1 yi ).

Proof. By changing variables xi → (xi − β)/(α − β), there is no loss of generality in assuming that α =1,β = 0. Then for any e1,...,em ≥ 0, we have

e1 em f(x1 ,...,xm )δyK ,L (dx)= f(x1,...,xm)δyK ,L (dx). f L f L i=1 yi i=1 yi Permuting the rows of an incidence matrix also gives another t-wise balanced design with the same parameters m, k1,...,kf ,λ. Thus, it suﬃces to show that 1 f(x)δyK ,L (dx)= f(x) f L |B| i=1 yi x∈X j ≤ ≤ ∈ for the monomials f(x)= i=1 xi, 1 j t. For this, we count the pairs (T ,B) V ×B ⊂ t ,T B in two ways: m λ = 1 t V ⊂ ∈B T ∈( ) T B t = 1 B∈B T ⊂B ∈ V T (t) f = 1 i=1 B∈Bi T ⊂B ∈ V T (t) f k = |B | i , i t i=1

t where regularity is used to show the ﬁrst identity. Thus, for f(x)= i=1 xi we have f |B | ki i t f(y)=λ = m . y∈X i=1 t This is further transformed to f m ki f |B | |B | m − t i · ki t i · m m = m ki − t i=1 ki t i=1 ki f |B | i = m f(x) i=1 ki x∈yL i |B| · = f(x)δyK ,L (dx). f L i=1 yi

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Remark 3.2. In a combinatorial framework (cf. [31]), some researchers regard t-wise balanced designs as cubature on “discrete spheres”. However, there are only a few publications where the regularity of designs is mentioned. Victoir seems to be the ﬁrst who employed combinatorial t-designs to reduce the size of cubature for usual continuous integrals. The following generalizes Theorem 2.12 (i) and motivates the study of regular t-wise balanced designs both in a combinatorial and analytic way. Theorem 3.3. Assume that there exists a regular q/2-wise balanced design with m points and bi blocks of size ki, i =1,...,e. Moreover, assume that there exists a cubature formula of degree q/2(or index q/2) for I´ of the form e b M w I´[f]=c i f(x) + i f(x) m b |xL| i=1 ki ∈ L i=2 i ∈ L x vki (α,β) x xi where b is the total number of blocks of the design and c is a positive number. Let X be the columns of a generalized incidence matrix with parameters α, β.Then c M w I´[f]= f(x)+ i f(x) b |xL| x∈X i=2 i ∈ L x xi is a cubature formula of degree q/2(or index q/2). The following proposition is often used in §4. Proposition 3.4. Assume there exists a t-(v, k, λ) design. Then: v−1 − { − } (t−1) − (i) There exists a regular t-(v 1, k, k 1 ,λ) design with λ k−1 blocks of size k 1 (t−1) v−1 (v−k)λ (t−1) and k k−1 blocks of size k. (t−1) (ii) Let X be the columns of an incidence matrix of the design given in (i), and let L L y = v (1, 0),y = v − (1, 0). Then for every f ∈P(y ∪ y ), 1 k 2 k 1 t 1 2 k−1 v−1 − − f(x)= t 1 k 1 f(x). λ v−1 ∈ L∪ L t−1 x∈X x y1 y2 Proof. (i) Let (V,B)beat-(v, k, λ) design, and let x ∈ V . We consider the incidence structure (V , B), where V = V \{x}, B = {B ∈B|x/∈ B}∪{B \{x}|x ∈ B ∈B}. Then (V , B) is a regular t-wise balanced design with the parameters determined by (2.4). Assertion (ii) follows from (i) and Proposition 3.1. We close this section with some remarks on regular t-wise balanced designs. First, as far as the authors know, there are only a few general results on the existence of regular t-wise balanced designs for t ≥ 3. Some examples are known, most of which are obtained trivially, like Proposition 3.4 (i). The second author and Reinhard Laue searched for regular 3-, 4- and 5-wise balanced designs with Discreta, a sophisticated program to compute designs, and found many designs with small parameters, some of which are summarized in Table 1. We believe that there will be further nontrivial regular t-wise balanced designs. How- ever, in this paper, such thorough discussions are omitted and left for future work. A natural problem is to ﬁnd a good bound for the number of blocks of a t-wise balanced design. Recently, Ziqing Xiang [37] derived a Fisher-type bound for regular

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Table 1. Some new regular t-wise balanced designs

Parameters Groups 3-(25, {6, 10}, 4) AGL(1, 25) 4-(27, {5, 8}, 5) ASL(3, 3) 5-(33, {6, 7}, 10) P ΓL(2, 32) 5-(33, {6, 8}, 20) P ΓL(2, 32) 5-(33, {6, 9}, 15) P ΓL(2, 32) 5-(33, {7, 10}, 42) P ΓL(2, 32) 5-(55, {6, 5}, 5) C2 × P ΓL(2, 27)

t-wise balanced designs. Namely, he showed that if there is a regular 2e-wise balanced design (V,B) with f distinct sizes of blocks, then f−1 |V | |B| ≥ . e − i i=0 This bound is sharp when t =2andf = 2, by a result of Woodal [36]. Moreover, for t =4andf = 2, a tight example can be constructed from the usual tight 4-design that corresponds to the Johnson scheme. Without regularity, no good bounds seem to be known.2

§4. Cubature arising from Victoir’s method and its generalization In this section many cubature formulas are constructed by Victoir’s method and its generalization formulated in §3. 4.1. Index-four cubature. There are many publications on the existence of index- four cubature in small-dimensional spaces that are not minimal but have few points; see, e.g., [26, 34]. In general-dimensional cases, however, it seems that explicit constructions of good cubature formulas are not sufficiently known.3 Therefore, the following theorem by Shatalov [32] is very important. Theorem 4.1. (i) [32, Theorem 4.4.9]. Assume that, for given m, n,andq, there exists a cubature formula of index q with n points on Sm−1. Then for any M ≥ m, there exists a cubature formula of index q with ((q +2)/2)M−mn points on SM−1. (ii) [32, Corollary 4.4.12]. There exists an index-four cubature formula on Sm−1 with n points when (4.1) m =22l + s, n =22l · 3s · (22l−1 +1),l≥ 1,s≥ 0, (4.2) m =2l +2+s, n =3s+1 · ((l +1)2 +1),lis a prime power,s≥ 0. Remark 4.2. For s = 0, Theorem 4.1 is a theorem of König [20]. The family (4.1) improves the upper-bound part of (2.1) if s is fixed and m is sufficiently large, or s =1, 2. A similar conclusion is valid for (4.2). We can construct cubature formulas in general-dimensional spaces that improve Shat- alov’s families.

2Eiichi Bannai kindly told us detailed information on bounds for regular t-wise balanced designs through email conversation. 3Oksana Shatalov and Yuan Xu kindly told us this information.

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Theorem 4.3. (i) Let l ≥ 2 and m be integers. Assume that 4l − 1 if 22l−1 ≤ m ≤ 22l; = 4l +1 if 22l

More Bm-invariant cubature formulas can be obtained systematically by using the Sobolev theorem. Proof of Theorem 4.3. (i) Take an OA(24l,m,2, 4) if 22l−1 ≤m≤22l,andanOA(24l+2,m,2, 4) if 22l

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opposite row-vectors of OA. The underlying symmetric cubature formulas were found by Victoir [35, Subsection 5.3]. (ii) Theorem 4.3 does not mention the exact number of points of the constructed cubature. When m =22l in Theorem 4.3 (i), the underlying OA is the Kerdock OA and no two distinct rows coincide. So, the resulting cubature has exactly 24l−1 +22l points, which is equivalent to König’s family. (iii) By Proposition 2.11, the L-invariant formula of Lemma 4.4 (i) is equivalent to the degree-five cubature of Stroud [34]. Moreover the formula in (ii) corresponds to Kürschák’s identity in number theory (see §6). 4.2. Index-six cubature. Shatalov [32, Theorem 4.7.20] compiled the known index-six cubature formulas with few points in small-dimensional spheres to get Table 2 (strictly speaking, a part of the original).

Table 2. Index-six cubature on Sm−1 with n points

No 1 2 3 4 5 6 7 8 9 10 11 12 13 m 3 4 5 6 7 8 9 10 11 16 17 18 23 n 11 23 41 63 113 120 480 1920 7680 2160 8640 34650 2300

Nos. 1, 2, 4 are (respectively) in [27, 11, 9].4 Nos. 3, 5 are in [34], and No. 6 is in [5]. To complete Table 2, Shatalov applied Theorem 4.1 (i) to one of the above formulas. For example, No. 7 has 4 times as many points as No. 6 does. According to Shatalov, Table 2 had not been updated so far, and the existence of general-dimensional index-six cubature formulas with few points is not fully known. Two families of general-dimensional cubature formulas that improve the upper-bound part of (2.1) can be given. Theorem 4.6. Let Q beaprimepowersuchthatQ ≡ 1(mod6),Q =25 .Letm ∈ {Q +1,Q},andletl be an integer with l ≥ 3, 22l−2

4The existence of a 23-point cubature formula of index 6 on S3 is not covered in [32, Theorem 4.7.20].

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Proof of Theorem 4.6. First, we consider the case where m = Q + 1. There exists a 3-(Q +1, (Q + 11)/6, (Q +5)(Q + 11)/72) design (cf. [19]), which has 3Q(Q + 1) blocks by (2.4). By Theorem 2.12 (i) and Lemma 4.7 (i), we obtain an index-three cubature for I´ with 1 + (Q +1)+3Q(Q + 1) points. By Proposition 2.11, this is equivalent to an L´-invariant cubature with 2Q+1 +2(Q +1)+2(Q+11)/6 · 3Q(Q + 1) points. By applying Theorem 2.12 (ii) to an OA(26l−1,Q+1, 2, 7) and an OA(26l−1, (Q + 11)/6, 2, 7) that are subarrays of the dual OA(26l−1, 22l, 2, 7) of the Delsarte–Goethals code (cf. [15, p. 103]), we obtain an index-six formula for I with at most 26l−1 · (1 + 3Q(Q +1))+2(Q +1) points. Note that the OA(26l−1, 22l, 2, 7) has central symmetry. In fact, the Delsarte– Goethals code can be constructed by applying the Gray-code mapping 0 → 00, 1 → 01, 2 → 11, 3 → 10 to linear, cyclic codes over Z4. Replacing 0, 1 ∈ F2 by ±1 implies the central symmetry of the OA. Thus, the result follows by Propositions 3.4 and 2.3. Similar arguments work when m = Q; replace the above L-invariant formula by that of Lemma 4.7 (ii). By Proposition 3.4 (ii) the above 3-design can be reduced to a regular 3-wise balanced design with Q points and 3Q(Q + 1) blocks. Using Theorem 3.3, we obtain an index-three cubature formula for I´ with 1 + 3Q(Q +1)+Q points. Now the claim follows by the same argument as in the case where m = Q +1.

Remark 4.8. The family of Theorem 4.6 has O(m5) points, improving the upper-bound part of (2.1). More general-dimensional index-six formulas with O(m5)pointsmaybe obtained by using known inﬁnite families of 3-designs [19]. Two more interesting cubature formulas can be given.

Example 4.9. The following is a 7-dimensional index-three cubature formula for I´: 1 1 (4.3) I´[f]= f(x)+ f(x). 140 √ √ 14 √ 3 L 3 L 3 L x∈v4( 28,0) ∪v3( 28,0) x∈v1( 112,0) A3-(8, 4, 1) design exists (cf. [19]), and a regular 3-(7, {4, 3}, 1) design with 7 blocks of sizes 4 and 3 also exists by Proposition 3.4 (i). Let X be the columns of an incidence matrix of the 3-wise balanced design. By Proposition 3.4 (ii), (4.4) f(x)=5 f(x) √ √ 3 L 3 L ∈ x∈v4( 28,0) ∪v3( 28,0) x X

for every f ∈P3. Hence, by (4.3), (4.4), and Proposition 2.11, we get the following index-six cubature for I: 1 1 1 (4.5) I[f]= f(x)+ f(x)+ f(x) 448 √ 224 √ 28 √ 6 L´ 6 L´ 3 B x∈( 28·X1) x∈( 28·X2) x∈v1( 112,0) m

where X1 = {x ∈ X | wt(x)=4},X2 = {x ∈ X | wt(x)=3}. Thisisreducedtoa 91-point formula of index 6 on S6, by Propositions 2.3 and 3.4.

Example 4.10. The following is a 9-dimensional index-three cubature formula for I´: 1 1 1 I´[f]= f(x)+ f(x)+ f(x). 3 630 √ √ 27 √ ∈ L 3 L 3 L 3 L x v9(1,0) x∈v4( 60,0) ∪v3( 60,0) x∈v1( 180,0) The existence of a 3-(10, 4, 1) design (cf. [19]) implies that of a regular 3-(9, {4, 3}, 1) design with 12 blocks of size 3 and 18 blocks of size 4. In the same way as in Example 4.9, a 457-point formula on S8 is obtained.

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Remark 4.11. (i) The formula No. 5 of Table 2 implies that N(7, 6) ≤ 113. Example 4.9 improves this to (4.6) N(7, 6) ≤ 91. The lower-bound part of (2.1) shows that 84 ≤ N(7, 6). The authors do not know of the existence of a cubature with fewer points than the 91-point formula on S6. It should also be noted that no spherical 84-point index-six cubature on S6 can exist, by Theorem 1 of [3]. (ii) The formula No. 7 of Table 2 implies that N(9, 6) ≤ 480. Example 4.10 improves this to (4.7) N(9, 6) ≤ 457. − − The fundamental√ roots of the group Bm are αi = ei ei+1 for i =1,...,m 1, Rm and αm = 2em,where√ e1,...,e√ m are the standard basis vectors in [4]. The corner vectors are vi =(1/ i,...,1/ i, 0,...,0) for i =1,...,m.WenotethatallBm-invariant cubature formulas of indices 4, 6 given in §4 consist of the orbits of the corner vectors. By Bajnok’s theorem, in order to ﬁnd higher-index spherical cubature formulas, we must take at least one orbit of points that are not corner vectors; see, e.g., [29] for a simple construction of higher-index formulas on spheres. A reﬁnement of Bajnok’s theorem is proved in the next section.

§5. The maximum strength of invariant Euclidean designs

We use the notations R, J, αi,vi,andX (G, J) defined in Subsection 2.3. Our aim in this section is to prove the following theorem. Theorem 5.1. Let G be a finite irreducible reflection group in Rm with m ≥ 2.Then there is no choice of R, J, and a weight w for which (X (G, J),w) is a Euclidean t-design of Rm in the following cases:

(i) t ≥ 6 if G = Am−1; (ii) t ≥ 8 if G = Bm,Dm; (iii) t ≥ 10 if G = E6; (iv) t ≥ 12 if G = F4,H3,E7; (v) t ≥ 16 if G = E8; (vi) t ≥ 24 if G = H4. The following lemma plays an important role in the proof of this theorem.

m−1 Lemma 5.2. Let G beasubgroupofO(m),andX = {x1,...,xM } a subset of S . { }mi Rm G Rm G Let fi,k k=1 be a basis of Harm2i( ) ,wheremi = dim(Harm 2i( ) ).LetVi be | ⊂ RX ∈ the space SpanR (fi,k(x1),...,fi,k(xM )) k =1,...,mi . Suppose there is v s i=1 Vi such that all entries of v are positive. Then there is no choice of radii ri and a M G weight w for which i=1 rixi ,w is a Euclidean 2s-design. Proof. Since X ⊂ Sm−1, we can write

s mi v = ai,k fi,k(x1),...,fi,k(xM ) i=1 k=1 s mi 2s−2i 2s−2i = ai,k x1 2 fi,k(x1),..., xn 2 fi,k(xM ) , i=1 k=1

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use REMARKS ON HILBERT IDENTITIES 629 s mi 2s−2i ∈ where the ai,k are real numbers. Let f(x):= i=1 k=1 ai,k x 2 fi,k(x). Then f 2j Rm G 2i+2j=2s,i≥1,j≥0 x 2 Harm2i( ) ,andf satisfies f(xi) > 0foreachi =1,...,M. g 2s g 2s It remains to observe that f(rixi )=ri f(xi )=ri f(xi) > 0fori =1,...,M and g ∈ G. Remark 5.3. Under our assumptions in Lemma 5.2, no subset of {rxg | g ∈ G, x ∈ } X, r > 0 can form a Euclidean 2s-design. In particular, for any subgroup H of G, M H i=1 rixi ,w is not a Euclidean 2s-design for any radii ri and weight w. The proof of Theorem 5.1 is divided into some cases. The following notation is used. For a finite irreducible reflection group G, vi denotes the corner vector normalized by | G| (vi,αi)=1,vi := vi/ (vi,vi), and Ni := vi .Letei be the column vector with the ith entry 1 and the others 0. Define 1 g g sym(f):= f(x ), (Sm)f := {g ∈ Sm | f(x )=f(x)} |(Sm)f | g∈Sm

for an m-variable polynomial f,whereSm is the symmetric group of m elements. Let 2 2 ··· 2 ≥ pi := x2 + x3 + + xi+1 for i 2. The polynomials hi in the following subsections are harmonic.

5.1. Group F4. Dynkin diagram: α1 α2 α3 α4 t t t t

Exponents:1, 5, 7, 11. Fundamental roots: −te − te − te + te α := te − te ,α := te − te ,α := te ,α := 1 2 3 4 . 1 1 2 2 2 3 3 4 4 2 Corner Vectors: t t t t t t t t t t v1 = e1 + e4,v2 = e1 + e2 +2 e4,v3 = e1 + e2 + e3 +3 e4,v4 =2e4.

Size of Orbit: N1 = 24, N2 = 96, N3 = 96, N4 = 24. Harmonic Molien series: 1 =1+t6 + t8 +2t12 + t14 + .... (1 − t6)(1 − t8)(1 − t12) G-invariant harmonic polynomials. 4 F4 For i =6, 8, 12, Harmi(R ) is spanned by the following polynomials. 1. Degree 6: 6 − 4 2 2 2 2 f6 := sym(x1) 5sym(x1x2)+30sym(x1x2x3). 2. Degree 8: 8 − 28 6 2 98 4 4 − 4 2 2 2 2 2 2 f8 := sym(x1) 3 sym(x1x2)+ 3 sym(x1x2) 28 sym(x1x2x3) + 504x1x2x3x4. 3. Degree 12: 12 − 10 2 8 4 f12,1 := sym(x1 ) 22 sym(x1 x2)+79sym(x1x2) 8 2 2 − 6 6 − 6 4 2 + 258 sym(x1x2x3) 116 sym(x1x2) 236 sym(x1x2x3) − 6 2 2 2 4 4 4 4 4 2 2 4392 sym(x1x2x3x4) + 570 sym(x1x2x3) + 3660 sym(x1x2x3x4), 12 − 10 2 133 8 4 f12,2 := sym(x1 ) 22 sym(x1 x2)+ 2 sym(x1x2) 591 8 2 2 − 157 6 6 − 1369 6 4 2 + 2 sym(x1x2x3) 2 sym(x1x2) 4 sym(x1x2x3) − 6 2 2 2 2265 4 4 4 6945 4 4 2 2 4167 sym(x1x2x3x4)+ 2 sym(x1x2x3)+ 2 sym(x1x2x3x4).

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Substitute vk for G-invariant harmonic polynomials. 1. Degree 6: − − 1 1 u6 := f6(v1),f6(v2),f6(v3),f6(v4)] = [ 1, 9 , 9 , 1 . 2. Degree 8: − 13 − 13 u8 := [f8(v1),f8(v2),f8(v3),f8(v4)] = 1, 27 , 27 , 1 . 3. Degree 12: u := [f (v ),f (v ),f (v ),f (v )] = 0, 128 , − 25 , 1 , 12,1 12,1 1 12,1 2 12,1 3 12,1 4 243 243 25 1751 u12,2 := [f12,2(v1),f12,2(v2),f12,2(v3),f12,2(v4)] = 128 , 3456 , 0, 1 .

Proposition 5.4. There is no choice of R, J,andw for which (X (F4,J),w) is a Eu- clidean 12-design.

Proof. Since − 25 7567 25 u12,1 +2u12,2 = 64 , 15552 , 243 , 1 , the claim follows from Lemma 5.2.

5.2. Group H3. Dynkin diagram: α1 α2 α3 t t t 5

Exponents:1, 5, 9. Fundamental roots: t t α1 := − e1 + e2, α := −te + te , 2 2 √ 3 √ √ √ √ √ (1 + 2+ 5 − 10)(te + te ) − (2 − 2+2 5+ 10)te α := 1 2 3 . 3 6 Corner Vectors: √ √ √ √ t t t −(3 2+ 10+8) e1 − (3 2+ 10 − 4)( e2 + e3) v1 = , √ √ 12 √ √ t t t −(3 2+ 10+2)(e1 + e2) − (3 2+ 10 − 4) e3 v2 = , √ √ 6 ( 2+ 10)(te + te + te ) v = − 1 2 3 . 3 4

Size of Orbit: N1 =12,N2 =30,N3 =20. Harmonic Molien series: 1 =1+t6 + t10 + t12 + t16 + t18 + t20 + .... (1 − t6)(1 − t10) G-invariant harmonic polynomials. 3 H3 For i =6, 10, 12, Harmi(R ) is spanned by the following polynomials. 1. Degree 6: √ f := 2 sym(x6)+21sym(x5x ) − 15 sym(x4x2)+21 10 sym(x4x x ) 6 1 √1 2 1 2√ 1 2 3 − − 3 3 − 3 2 2 2 2 (70 7 10) sym(x1x2) 21 10 sym(x1x2x3) + 180x1x2x3. 2. Degree 10: g f10 := h10(x ),

g∈H3

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where 10 − 8 6 2 − 4 3 2 4 − 5 h10(x) := 256x1 5760x1p2 + 20160x1p2 16800x1p2 + 3150x1p2 63p2. 3. Degree 12: g f12 := h12(x ), ∈ where g H3 12 − 10 8 2 h12(x) := 1024x1 33792x1 p2 + 190080x1p2 − 6 3 4 4 − 2 5 6 295680x1p2 + 138600x1p2 16632x1p2 + 231p2.

Substitute vk for G-invariant harmonic polynomials. 1. Degree 6: √ √ √ 14 10−4 −7 10+2 −14 10+4 u6 := [f6(v1),f6(v2),f6(v3)] = 5 , 8 , 9 . 2. Degree 10: u10 := [f10(v1),f10(v2),f10(v3)] √ √ √ − 43124224 10+49637120 8422700 10+9694750 − 1078105600 10+1240928000 = 98415 , 19683 , 1594323 . 3. Degree 12: u12 := [f12(v1),f12(v2),f12(v3)] √ √ √ 191679488 10−6897476096 10856846 10−390677357 −191679488 10+6897476096 = 492075 , 39366 , 14348907 .

Proposition 5.5. There is no choice of R, J,andw for which (X (H3,J),w) is a Eu- clidean 12-design.

Proof. There is u ∈ SpanR{u6,u10,u12} all whose entries are positive, because the vectors u6, u10, u12 are linearly independent. The result follows by Lemma 5.2.

5.3. Group H4. Dynkin diagram: α1 α2 α3 α4 t t t t 5 Exponents:1, 11, 19, 29. Fundamental roots: √ te + te + te + 5 te α := − te + te ,α := − te + te ,α := − te + te ,α := 1 2 3 4 . 1 1 2 2 2 3 3 3 4 4 2 Corner Vectors: √ √ t t t t ( 5 − 1) e1 +( 5+3)( e2 + e3 − e4) v1 = , √ 4 √ t t t t ( 5+1)(e1 + e2)+( 5+3)(e3 − e4) v2 = , √ 2 √ t t t t (3 5+5)(e1 + e2 + e3) − 3( 5+3) e4 v3 = , √ 4 ( 5+3)(te + te + te − te ) v = 1 2 3 4 . 4 2 Size of Orbit: N1 = 120,N2 = 720,N3 = 1200,N4 = 600. Harmonic Molien series: 1 =1+t12 + t20 + t24 + t30 + .... (1 − t12)(1 − t20)(1 − t30)

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G-invariant harmonic polynomials. 4 H For i =12, 20, 24, Harmi(R ) 4 is spanned by the following polynomials. 1. Degree 12: g f12 := h12(x ), ∈ where g H4 12 − 10 8 2 − 6 3 4 4 − 2 5 6 h12(x):=13x1 286x1 p3 + 1287x1p3 1716x1p3 + 715x1p3 78x1p3 + p3. 2. Degree 20: g f20 := h20(x ), ∈ where g H4 20 − 18 16 2 − 14 3 12 4 h20(x):=21x1 1330x1 p3 + 20349x1 p3 116280x1 p3 + 293930x1 p3 − 10 5 8 6 − 6 7 4 8 − 2 9 10 352716x1 p3 + 203490x1p3 54264x1p3 + 5985x1p3 210x1p3 + p3 . 3. Degree 24: g f24 := h24(x ), ∈ where g H4 24 − 22 10626 20 2 − 18 3 16 4 h24(x):=x1 92x1 p3 + 5 x1 p3 19228x1 p3 + 81719x1 p3 − 14 5 12 6 − 653752 10 7 8 8 178296x1 p3 + 208012x1 p3 5 x1 p3 + 43263x1p3 − 6 9 4 10 − 2 11 1 12 7084x1p3 + 506x1p3 12x1p3 + 25 p3 .

Substitute vk for G-invariant harmonic polynomials. 1. Degree 12: − 32500 5625 u12 := [f12(v1),f12(v2),f12(v3),f12(v4)] = 4500, 540, 27 , 4 . 2. Degree 20: − 58869 4035425 216225 u20 := [f20(v1),f20(v2),f20(v3),f20(v4)] = 6975, 25 , 2187 , 64 . 3. Degree 24: − 2367 − 4689027 416329 622521 u24 := [f24(v1),f24(v2),f24(v3),f24(v4)] = 16 , 50000 , 104976 , 16384 .

Proposition 5.6. There is no choice of R, J,andw for which (X (H4,J),w) is a Eu- clidean 24-design.

Proof. Since − 91305 2293281 30201755 18338985 u20 30u24 = 8 , 5000 , 17496 , 8192 , this proposition follows from Lemma 5.2.

5.4. Group E6. Dynkin diagram: α1 α2 α3 α4 α5 t t t t t

t α6 Exponents:1, 4, 5, 7, 8, 11. Fundamental roots: α := te − te ,α := te − te ,α := te − te ,α := te − te , 1 1 2 2 2 √3 3 3 4 4 4√ 5 (−3+ 3)(te + te + te )+(3+ 3)(te + te + te ) α := te − te ,α := 1 2 3 4 5 6 . 5 5 6 6 6

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Corner Vectors: √ √ t t t t t t ( 3+5) e1 +( 3 − 1)( e2 + e3 + e4 + e5 + e6) v1 = , √ √6 t t t t t t ( 3+2)( e1 + e2)+( 3 − 1)( e3 + e4 + e5 + e6) v2 = , √ 3 √ t t t t t t ( 3+1)( e1 + e2 + e3)+( 3 − 1)( e4 + e5 + e6) v3 = , √ 2 √ t t t t t t ( 3+1)( e1 + e2 + e3 + e4)+( 3 − 2)( e5 + e6) v4 = , √ 3 √ t t t t t t ( 3+1)( e1 + e2 + e3 + e4 + e5)+( 3 − 5) e6 v5 = , √ 6 3(te + te + te + te + te + te ) v = 1 2 3 4 5 6 . 6 3

Size of Orbit: N1 =27,N2 = 216,N3 = 720,N4 = 216,N5 =27,N6 = 72. Harmonic Molien series: 1 =1+t5 + t6 + t8 + t9 + t10 + .... (1 − t5)(1 − t6)(1 − t8)(1 − t9)(1 − t12) G-invariant harmonic polynomials. 6 E6 For i =5, 6, 8, 9, 10, Harmi(R ) is spanned by the following polynomials. 1. Degree 5: 5 4 − 3 2 3 f5 := sym(x1)+sym(x1x2) 2sym(x1x2)+sym(x1x2x3) − 2 3sym(x1x2x3x4)+24sym(x1x2x3x4x5). 2. Degree 6: 6 3 5 − 4 2 15 4 f6 := sym(x1)+ 2 sym(x1x2)+ 3sym(x1x2)+ 14 sym(x1x2x3) 5 3 3 − 30 3 2 30 3 2 2 2 + 7 sym(x1x2) 7 sym(x1x2x3)+ 7 sym(x1x2x3x4)+9sym(x1x2x3) 45 2 2 − 180 2 180 + 7 sym(x1x2x3x4) 7 sym(x1x2x3x4x5)+ 7 x1x2x3x4x5x6.

3. Degree 8: g f8 := h8(x ), ∈ where g E6 8 − 28 6 4 2 − 4 2 3 1 4 h8(x):=x1 5 x1p5 +6x1p5 3 x1p5 + 33 p5. 4. Degree 9: g f9 := h9(x ), ∈ where g E6 9 − 36 7 2 126 5 4 − 4 3 2 h9(x):=sym(x1) 5 sym(x1x2)+ 5 sym(x1x2) 63 sym(x1x2x3) 4 2 2 3 2 2 2 − 2 2 2 2 +63sym(x1x2x3x4) + 252 sym(x1x2x3x4) 945 sym(x1x2x3x4x5).

5. Degree 10: g f10 := h10(x ), ∈ where g E6 10 − 8 6 2 − 4 3 15 2 4 − 3 5 h10(x):=x1 9x1p5 +18x1p5 10x1p5 + 11 x1p5 143 p5.

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Substitute vk for G-invariant harmonic polynomials. 1. Degree 5: √ √ √ √ 3 3 6 30 − 6 30 − 3 3 u5 := [f5(v1),f5(v2),f5(v3),f5(v4),f5(v5),f5(v6)] = 4 , 125 , 0, 125 , 4 , 0 .

2. Degree 6: 81 − 81 − 9 − 81 81 − 27 u6 := [f6(v1),f6(v2),f6(v3),f6(v4),f6(v5),f6(v6)] = 56 , 700 , 28 , 700 , 56 , 28 .

3. Degree 8:

u := [f (v ),f (v ),f (v ),f (v ),f (v ),f (v )] 8 8 1 8 2 8 3 8 4 8 5 8 6 − 6784 − 640 − 6784 3200 = 800, 25 , 9 , 25 , 800, 3 .

4. Degree 9:

u9 := [f9(v1),f9(v2),f9(v3),f9(v4),f9(v5),f9(v6)] √ √ √ √ − 185024 30 185024 30 − = 2065 3, 625 , 0, 625 , 2065 3, 0 .

5. Degree 10:

u := [f (v ),f (v ),f (v ),f (v ),f (v ),f (v )] 10 10 1 10 2 10 3 10 4 10 5 10 6 11520 423936 51200 423936 11520 − 10240 = 13 , 1625 , 351 , 1625 , 13 , 39 .

Proposition 5.7. There is no choice of R, J,andw for which (X (E6,J),w) is a Eu- clidean 10-design.

Proof. Since 11745 13527 387 13527 11745 621 u10 + u8 = 2816 , 220000 , 1760 , 220000 , 2816 , 352 , this proposition follows from Lemma 5.2.

5.5. Group E7. Dynkin diagram: α1 α2 α3 α4 α5 α6 t t t t t t

t α7 Exponents:1, 5, 7, 9, 11, 13, 17. Fundamental roots:

α := te − te ,α:= te − te ,α:= te − te ,α:= te − te ,α:= te − te , 1 1 2 2 2 √3 3 3 4 4 √4 5 5 5 6 (−4+ 2)(te + te + te )+(3+ 2)(te + te + te + te ) α := te − te ,α:= 1 2 3 4 5 6 7 . 6 6 7 7 7

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Corner Vectors: √ √ t t t t t t t (6 + 2 2) e1 +(−1+2 2)( e2 + e3 + e4 + e5 + e6 + e7) v1 = , √ 7√ t t t t t t t (5 + 4 2)( e1 + e2)+(−2+4 2)( e3 + e4 + e5 + e6 + e7) v2 = , √ 7 √ t t t t t t t (4 + 6 2)( e1 + e2 + e3)+(−3+6 2)( e4 + e5 + e6 + e7) v3 = , √ 7 √ t t t t t t t (6 + 9 2)( e1 + e2 + e3 + e4)+(−8+9 2)( e5 + e6 + e7) v4 = , √ 14 √ t t t t t t t (2 + 3 2)( e1 + e2 + e3 + e4 + e5)+(−5+3 2)( e6 + e7) v5 = , √ 7 √ (−2 − 3 2)(te + te + te + te + te + te )+(12− 3 2)te v = 1 2 3 4 5 6 7 , 6 14 t t t t t t t e1 + e2 + e3 + e4 + e5 + e6 + e7 v7 = √ . 2 Size of Orbit:

N1 = 126,N2 = 2016,N3 = 10080,N4 = 4032,N5 = 756,N6 =56,N7 = 576. Harmonic Molien series: 1 =1+t6 + t8 + t10 +2t12 + .... (1 − t6)(1 − t8)(1 − t10)(1 − t12)(1 − t14)(1 − t18) G-invariant harmonic polynomials. 7 E For i =6, 8, 10, 12, Harmi(R ) 7 is spanned by the following polynomials. 1. Degree 6: g f6 := h6(x ), ∈ where g E7 6 − 4 2 2 − 3 h6(x):=32x1 80x1p6 +30x1p6 p6. 2. Degree 8: g f8 := h8(x ), ∈ where g E7 8 − 6 4 2 − 2 3 4 h8(x) := 384x1 1792x1p6 + 1680x1p6 336x1p6 +7p6. 3. Degree 10: g f10 := h10(x ), ∈ where g E7 10 − 8 6 2 − 4 3 2 4 − 5 h10(x) := 256x1 1920x1p6 + 3360x1p6 1680x1p6 + 210x1p6 3p6. 4. Degree 12: g g f12,1 := h12,1(x ),f12,2 := h12,2(x ), g∈E g∈E where 7 7 12 − 10 8 2 h12,1(x) := 4096x1 45056x1 p6 + 126720x1p6 − 118272x6p3 + 36960x4p4 − 3168x2p5 +33p6, 1 6 1 6 1 6 6 10 − 8 6 2 − 4 3 2 4 − 5 h12,2(x):=x1x2 2048x1 14080x1p6 + 25344x1p6 14784x1p6 + 2640x1p6 99p6 .

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Substitute vk for G-invariant harmonic polynomials. 1. Degree 6:

u6 := [f6(v1),f6(v2),f6(v3),f6(v4),f6(v5),f6(v6),f6(v7)] √ √ √ −7700659200+9488793600 2 −427814400+527155200 2 −1818211200+2240409600 2 = 16807 , 2401 , 16807 , √ √ √ −547602432+674758656 2 2887747200−3558297600 2 20535091200−25303449600 2 16807 , 16807 , 16807 , √ −123210547200+151820697600 2 823543 .

2. Degree 8:

u8 := [f8(v1),f8(v2),f8(v3),f8(v4),f8(v5),f8(v6),f8(v7)] √ √ 6579988992000−5480856576000 2 731109888000−608984064000 2 = 823543 , 823543 , √ √ −3527605209600+2938348108800 2 −3134999199744+2611323666432 2 823543 , 823543 , √ √ −1809496972800+1507235558400 2 3509327462400−2923123507200 2 823543 , 823543 , √ −115807806259200+96463075737600 2 40353607 .

3. Degree 10:

u10 := [f10(v1),f10(v2),f10(v3),f10(v4),f10(v5),f10(v6),f10(v7)] √ √ −6428624451840+415928908800 2 357145802880−23107161600 2 = 5764801 , 823543 , √ √ 2388412556760−154529143200 2 −73143460429824+4732346695680 2 5764801 , 720600125 , √ √ −7433097022440+480917800800 2 30476441845760−1971811123200 2 5764801 , 5764801 , √ 3291455719342080−212955601305600 2 1977326743 .

4. Degree 12:

u12,1 := [f12,1(v1),f12,1(v2),f12,1(v3),f12,1(v4),f12,1(v5),f12,1(v6),f12,1(v7)] √ √ 27363005574796800+17942314142016000 2 −760132890073600−689327933184000 2 = 1977326743 , 282475249 , √ √ −513174301527400−792264524693400 2 −14026148038967296−72141536776421376 2 1977326743 , 49433168575 , √ √ 3931481294451000−4960153279164600 2 −12979679661260800−37832882828083200 2 1977326743 , 1977326743 , √ 249757080640811827200+284898146732782387200 2 33232930569601 . u12,2 := [f12,2(v1),f12,2(v2),f12,2(v3),f12,2(v4),f12,2(v5),f12,2(v6),f12,2(v7)] √ √ −2419675164360000−1489162193640000 2 113867977056000+18727597152000 2 = 1977326743 , 282475249 , √ √ 156757191916575−26126480038725 2 16622260339703808−6701937797136384 2 1977326743 , 49433168575 , √ √ 1494452243214675−1107985853945025 2 8313279170969600−2771226582220800 2 1977326743 , 1977326743 , √ −51686387833407897600+773622407142604800 2 33232930569601 .

Proposition 5.8. There is no choice of R, J,andw for which (X (E7,J),w) is a Eu- clidean 12-design.

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Proof. Since

− 2u12,1 − 25u12,2 + u10 =[2.86443 × 106, 256489., 513956., 989894., 2.86352 × 106, 1.64917 × 107, 293023.], this proposition follows from Lemma 5.2.

5.6. Group E8. Dynkin diagram: α1 α2 α3 α4 α5 α6 α7 t t t t t t t

t α8 Exponents:1, 7, 11, 13, 17, 19, 23, 29. Fundamental roots: t t t t t t t t t t α1 := e1 − e2,α2 := e2 − e3,α3 := e3 − e4,α4 := e4 − e5,α5 := e5 − e6, −te − te − te + te + te + te + te + te α := te − te ,α:= te − te ,α:= 1 2 3 4 5 6 7 8 . 6 6 7 7 7 8 8 2 Corner Vectors: 3 te + te + te + te + te + te + te + te v = 1 2 3 4 5 6 7 8 , 1 2 t t t t t t t t v2 =2 e1 +2 e2 + e3 + e4 + e5 + e6 + e7 + e8, 5 te +5te +5te +3te +3te +3te +3te +3te v = 1 2 3 4 5 6 7 8 , 3 2 t t t t t t t t v4 =2 e1 +2 e2 +2 e3 +2 e4 + e5 + e6 + e7 + e8, 3 te +3te +3te +3te +3te + te + te + te v = 1 2 3 4 5 6 7 8 , 5 2 t t t t t t v6 = − e1 − e2 − e3 − e4 − e5 − e6, −te − te − te − te − te − te − te + te v = 1 2 3 4 5 6 7 8 , 7 2 t t t t t t t t v8 = e1 + e2 + e3 + e4 + e5 + e6 + e7 + e8. Size of Orbit:

N1 = 2160,N2 = 69120,N3 = 483840,N4 = 241920,

N5 = 60480,N6 = 6720,N7 = 240,N8 = 17280. Harmonic Molien series: 1 (1 − t8)(1 − t12)(1 − t14)(1 − t18)(1 − t20)(1 − t24)(1 − t30) =1+t8 + t12 + t14 + t16 + t18 +2t20 + .... G-invariant harmonic polynomials. 8 E8 For i =8, 12, 14, 16, Harm8(R ) is spanned by the following. 1. Degree 8: g f8 := h8(x ), ∈ where g E8 8 − 6 4 2 − 2 3 4 h8(x) := 429x1 1716x1p7 + 1430x1p7 260x1p7 +5p7.

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2. Degree 12: g f12 := h12(x ), ∈ where g E8 12 − 10 8 2 − 6 3 4 4 − 2 5 6 h12(x) := 1547x1 14586x1 p7 + 36465x1p7 30940x1p7 + 8925x1p7 714x1p7 +7p7. 3. Degree 14: g f14 := h14(x ), ∈ where g E8 14 − 12 10 2 − 8 3 h14(x) := 969x1 12597x1 p7 + 46189x1 p7 62985x1p7 6 4 − 4 5 2 6 − 7 + 33915x1p7 6783x1p7 + 399x1p7 3p7. 4. Degree 16: g f16 := h16(x ), ∈ where g E8 16 − 14 12 2 − 10 3 h16(x) := 6783x1 116280x1 p7 + 587860x1 p7 1175720x1 p7 8 4 − 6 5 4 6 − 2 7 8 + 1017450x1p7 379848x1p7 + 55860x1p7 2520x1p7 +15p7.

Substitute vk for G-invariant harmonic polynomials. 1. Degree 8: u8 := [f8(v1),f8(v2),f8(v3),f8(v4),f8(v5),f8(v6),f8(v7),f8(v8)] 4926873600 = [174182400, 49 , 82059264, 62705664, 19353600, −116121600, −1045094400, 97977600]. 2. Degree 12: u12 := [f12(v1),f12(v2),f12(v3),f12(v4),f12(v5),f12(v6),f12(v7),f12(v8)] 15655887360 14950365696 − 2608490304 = [1680315840, 49 , 125 , 125 , − 275607360, −734952960, 4480842240, 148777965]. 3. Degree 14: u14 := [f14(v1),f14(v2),f14(v3),f14(v4),f14(v5),f14(v6),f14(v7),f14(v8)] − 567924825600 − 2009165312 − 671799744 = [1207483200, 16807 , 15 , 5 , − 253422400 − − 3 , 184307200, 2634508800, 293294925]. 4. Degree 16: u16 := [f16(v1),f16(v2),f16(v3),f16(v4),f16(v5),f16(v6),f16(v7),f16(v8)] − 393199971840 3287394820358656 36512571016971 = [1490121360, 2401 , 1265625 , 62500 , 1232569520 − 9749511135 3 , 2075906560, 7529034240, 16 ].

Proposition 5.9. There is no choice of R, J,andw for which (X (E8,J),w) is a Eu- clidean 16-design. Proof. Since − 9691313402880 4098709695302656 59096571112971 u16 3u14 +2u12 = [1228303440, 16807 , 1265625 , 62500 , 339192560 9089540145 3 , 53079040, 24394245120, 16 ], this proposition follows from Lemma 5.2.

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Now, we are ready to complete the proof of Theorem 5.1.

Proof of Theorem 5.1. Case (1) is in Theorem 2.10, and cases (2), (3) in [24]. Thus, the theorem follows from Propositions 5.4–5.9.

The following result, together with Theorem 5.1, determines the maximal degree of the spherical cubature formulas (X (G, J),w) for all irreducible reﬂection groups G. Theorem 5.10. (i) An F4-invariant cubature of degree 11 that consists of the orbits of corner vectors is classiﬁed by: − − − 13 960w4 3( 1+192w4) 3(1 120w4) 1 ≤ ≤ 1 w1 = 960 ,w2 = 256 ,w3 = 160 , 192 w4 120 .

(ii) An H3-invariant cubature of degree 11 that consists of the orbits of corner vectors is classiﬁed by: 125 64 27 w1 = 5544 ,w2 = 3465 ,w3 = 3080 .

(iii) An H4-invariant cubature of degree 23 that consists of the orbits of corner vectors is classiﬁed by:

368−9625w4 125(16+5625w4) w1 = 315392 ,w2 = 2359296 , − − 6561(16 51975w4) ≤ ≤ 16 w3 = 504627200 , 0 w4 51975 .

(iv) An E6-invariant cubature of degree 9 that consists of the orbits of corner vectors is classiﬁed by: − 2(1 96w6) 125(1+1200w6) 1 − 9w6 w1 = 729 ,w2 = 186624 ,w3 = 1280 16 , − 125(1+1200w6) 2(1 96w6) ≤ ≤ 1 w4 = 186624 ,w5 = 729 , 0 w6 720 .

(v) An E7-invariant cubature of degree 11 that consists of the orbits of corner vectors is classiﬁed by the following two types of weights: − − 4( 296924467+966078461040w2 +107900687895w3+95875084800w7) (1) w1 = 610410794301 , − − 625( 945994+3215011030w2 +24066363475w3+1769169600w7) w4 = 4340698981696 ,

8(34900936+247702641648w2 +1231161574335w3+182083866624w7) w5 = 1831232382903 , − − 27( 32430307+60983896974w2 +30607311735w3+25518620160w7) w6 = 542587372712 , − ≤ ≤−2401( 394+1339030w2+10023475w3) 0 w7 1769169600 , − ≤ − 2( 197+669515w2) ≤ ≤ 197 0 w3 < 10023475 , 0 w2 669515 . − − − 4( 211+686070w2) − 2( 197+669515w2) (2) w1 = 440055 ,w3 = 10023475 , − 16(1231+1230075w2) − 351( 71+129360w2) w5 = 54126765 ,w6 = 16037560 , ≤ ≤ 197 0 w2 669515 ,w4 =0,w7 =0.

(vi) An E8-invariant cubature of degree 15 that consists of the orbits of corner vectors is classiﬁed by the nonnegative solutions wi of the system of equations 8 t t t (5.1) u8 v =0,u12 v =0,u14 v =0, Niwi =1, i=1

where v =(N1w1,...,N8w8),andtheui, Ni are as deﬁned in Subsection 5.6.The precise solutions of (5.1) are referred to the Appendix.

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Remark 5.11. The H3-invariant cubature of Theorem 5.10 (i) was constructed by Goeth- als and Seidel [9, p. 214] who found, moreover, a spherical cubature of degree 15 by taking the orbits of v1,v2,v3, plus one more orbit; for example, see [12] for further information on the existence of three-dimensional spherical cubature formulas. It is also interesting to note that the formula given in Theorem 5.10 (vi) is equivalent to a 26400-point cubature of degree 15 which comes from shells of the Korkin–Zorotalev lattice [13]. In [9, p. 214], Goethals and Seidel found a spherical cubature of degree 19 that consists of the H4-orbits of the zeros of an invariant harmonic homogeneous polynomial of degree 12. Salihov [28] found another H4-invariant cubature of degree 19 by taking the union of the 120-cell and the 600-cell. Motivated by this, the authors searched three and four H4-orbits of the corner vectors, and found the higher-degree cubature of Theorem 5.10 (ii).

§6. Hilbert identities and cubature formulas As was explained in §2, a cubature formula of index q on Sm−1 with n points exists m if and only if there are n vectors r1,...,rn ∈ R such that n q q (6.1) x, ri = x, x 2 i=1 ∈ Rm m 2 q/2 for every x . Identity (6.1) yields a representation of ( i=1 xi ) as a sum of qth powers of real linear forms with positive real coefficients. Such a representation is called a Hilbert identity [25]. Various aesthetic meanings of Hilbert identities were discussed in a famous paper by Reznick, see [27]. Many Hilbert identities can be obtained with the help of the cubature formulas that were constructed in §4and§5. In particular, some of the resulting identities involve sums of qth powers of rational linear forms with positive rational coefficients. Such rational representations were used not only in studying Waring’s problem [6, pp. 717–725], but also in the work of Schmid on real holomorphy rings [30]. An aesthetic meaning of 5 rational representations would be stated as follows. We would take all coefficients {ai} that appear in a formula, and consider the field created by adjoining them, and then look at its dimension [Q({ai}):Q]. With this measure, the “best formulas” would only involve rationals, and the minimum value occurs if the coefficients are already in Q. It is well known (going back to Hilbert [16]) that q (q − 1)!!(m − 2)!! (6.2) y1ρ (dy)= − . Sm−1 (m + q 2)!! This is certainly a rational number. All cubature√ formulas√ given in §4 have rational weights, and points in orbits are of the form ( q a,..., q a, 0,...,0)Bm with rational a. Thus, by Proposition 2.3, we can obtain many rational representations. For example, the 91-point cubature of Example 4.9 is translated into the following rational representation, which Reznick [27] was not able to find. Theorem 6.1. 7 3 2 ± ± ± 6 120 xi = (xi xi+2 xi+3 xi+4) (6.3) i=1 56 6 6 +2 (xi ± xi+2 ± xi+3) + (2xi) 28 7 where the indices on the right are taken cyclic modulo 7 and all possible combinations of signs occur in summation.

5This was suggested by Bruce Reznick through email conversation.

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Remark 6.2. Reznick [27, p. 112] translated an index-six cubature on S6 found by Stroud in 1967 into the following beautiful representation: 7 3 2 6 ± 6 ±···± 6 (6.4) 960 xi =2 (2xi) + (2xi 2xj) + (x1 x7) , i=1 7 · 7 26 2 (2)

where on the right all possible combinations of signs and pairs of the 7 variables x1,...,x7 occur in the second summation. Identity (6.3) improves Reznick’s representation. Na- mely, (6.3) has fewer number of sixth powers than (6.4). More rational representations are available. For example, look at the following Kürs- chák’s representation: 3k 3k+1 2 2k x2 = (x ± x ±···±x )4 k i i1 i2 ik+1 i=1 where on the right all possible combinations of signs and (k+1)-subsets of the 3k+1 variables x1,...,x3k+1 occur [6, p. 723]. This corresponds to the cubature of Lemma 4.4 (ii), which, by Theorem 4.3, reduces to many rational representations involving much fewer number of fourth powers. We give yet another interesting Hilbert identity, though it is not always rational. Theorem 6.3. 4 5 2 1 10 1 ± ± ± 10 xi = 2520 (2xi) + 2520 (x1 x2 x3 x4) i=1 4 8 1−120a ± ± ± 10 1−120a ± ± 10 (6.5) + 272160 (3xi xj xk xl) + 272160 (2xi 2xj 2xk) 32 16 192a−1 ± ± 10 12−960a ± 10 + 68040 (2xi xj xk) + 630 (xi xj ) , 48 12 1 ≤ ≤ 1 where 192 a 120 .Inparticular,ifa is rational, then so is the corresponding identity. Proof. The cubature of Theorem 5.10 (1) is centrally symmetric, which reduces to the half-size formula of index 10. The result then follows by (6.1) and (6.2). Identity (6.5) unifies the following well-known identity by I. Schur (cf. [6, p. 721]). Corollary 6.4. 4 5 2 10 ± ± ± 10 22680 xi =9 (2xi) +9 (x1 x2 x3 x4) (6.6) i=1 4 8 10 10 + (2xi ± xj ± xk) + 180 (xi ± xj ) . 48 12 Proof. Take a =1/120 in (6.5). Remark 6.5. Some classical identities as such by Lucus (1876) and Liouville (1859), are often picked up for an introduction in the study of Hilbert identities [6]. It is well known (see, e.g., [14, 27]) that Liouville’s and Lucas’s identities are closely related to each other by a linear change and provide essentially the same cubature on S3. The Hurwitz identity 4 4 2 8 ± ± ± 8 5040 xi =6 (2xi) +6 (x1 x2 x3 x4) i=1 4 8 8 8 + (2xi ± xj ± xk) +60 (xi ± xj ) 48 12

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is also well known [6, p. 721]. It is interesting to note that Hurwitz’s and Schur’s identities are the same in terms of spherical cubature, i.e., the corresponding formulas have the same weights and points. In [14, 27], this observation was not remarked, though the relationship between Liouville’s and Lucas’s identities was mentioned. The story so far implies how powerful the cubature approach is to construct Hilbert identities. In turn, we look at an advantage of translating spherical cubature into Hilbert identities. ≥ m 2 4 Theorem 6.6. Let m 2 be an integer. Then i=1 xi cannot be represented as an 8 R-linear combination of (a1x1 + ···+ amxm) with ai ∈{0, −1, 1}. 6 2 4 4 n 2 4 Proof. The ratio of the coeﬃcients of x1x2 and x1x2 is (2 : 3) in ( i=1 xi ) . But it is 8 (2 : 5) in any form (a1x1 + ···+ anxn) with ai ∈{0, ±1},0∈{/ a1,a2}.

Corollary 6.7. Let m ≥ 2,andletG be a subgroup of Bm. Then there exists no G-invariant Euclidean 8-design of Rm that consists of the orbits of the form (1,...,1, 0,...,0)G. Proof. Restricting (2.3) to homogeneous polynomials of degree 8 implies the existence of a cubature formula of index 8 on Sm−1, by suitably rescaling points and weights. The result then follows by Theorem 6.6. A variation of Corollary 6.7 is valid for all irreducible reflection groups. Namely, Theorem 5.1 can be proved even if each irreducible reflection group is replaced by its subgroup. Remark 6.8. (i) Corollary 6.7 is the Bajnok theorem for G = Bm, and case (3) of Theorem 5.1 for G = Dm. It is also interesting to note that Theorem 6.6 states that the Bajnok theorem is valid even if negative coefficients are allowed. (ii) To prove Theorem 2.10, Bajnok used the Sobolev theorem implicitly. The approach based on the Sobolev theorem is of theoretic interest, but it basically requires tedious calculations on invariant harmonic homogeneous polynomials. In summary, the original proof of Bajnok requires a few pages [1, §2 and Proposition 15] and seems to be involved. Whereas, the present proof is short and simple, because it only involves elementary counting techniques. The Bajnok theorem is well known in algebra and combinatorics; however, it is not fully recognized in numerical analysis, though it can be used to determine the maximal degree of a symmetric cubature on the simplex [39], which is traditionally studied in the context of numerical analysis.6 The authors expect that the new proof will make researchers in many fields more familiar with the Bajnok theorem. Acknowledgment. This work started when the second author stayed at the Depart- ment of Mathematics of the University of Oregon from April to June in 2011. He grate- fully acknowledges the hospitality of this institution and the cooperation with Yuan Xu and many other staffs. The authors also thank Eiichi Bannai, Reinhard Laue, Sanpei Kageyama, and Oksana Shatalov for fruitful discussions about regular t-wise balanced designs and index-type cubature formulas. After an earlier version of this paper was written, the second author emailed Bruce Reznick and Koichi Kawada to discuss the content of §5and§6. They were really patient in giving us some elementary courses in the subject and many valuable comments and suggestions; the resulting revision extensively improved the previous version.

6The second author learned this fact form Yuan Xu. In [29], we proved a variation of the Bajnok theorem for cubature formulas on the simplex, particularly intended for researchers in numerical analysis.

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Appendix A. Classification of E8-invariant cubature

An E8-invariant cubature of degree 15 that consists of the orbits of corner vectors is classiﬁed by the following 27 types of weights:

23 − 4288512w2 − 258048w3 − 70224w4 − 15w8 w1 = 504000 823543 15625 15625 128 , 3 − 1244160w2 − 171008w3 − 79704w4 − 243w8 w5 = 224000 823543 15625 15625 512 , 9 4193208w2 507384w3 180792w4 3645w8 w6 = 896000 + 823543 + 15625 + 15625 + 2048 , 67 − 2465280w2 − 290304w3 − 94752w4 − 603w8 w7 = 672000 823543 15625 15625 512 , and − ≤ ≤ 12588443 ≤ 44118375 4976640000000w2 (1) w4 =0,0 w2 1449551462400 ,0 w3 < 36053104984064 , − − ≤ ≤ 88236750 9953280000000w2 72106209968128w3 0 w8 3126889828125 , − ≤ ≤ 12588443 44118375 4976640000000w2 (2) w4 =0,0 w2 1449551462400 , w3 = 36053104984064 , w8 =0, 12588443 117649 (3) w4 =0, 1449551462400

117649 2705927 338240875−38596608000000w2 (10) w4 =0, 13436928000

125 62942215+7929414597888w4 (14) 0

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125 62942215+7929414597888w4 (15) 0

125 14706125−1123879249584w4 (19) 0

125 14706125−1123879249584w4 (20) 0

125 14706125−1123879249584w4 (21) 0

125 338240875−33311510572032w4 (24) 0

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Department of Mathematics, Aichi University of Education, Igaya-cho, Kariya-city 448- 8542, Japan E-mail address: [email protected] Graduate School of Information Sciences, Nagoya University, Chikusa-ku, Nagoya 464- 8601. Japan E-mail address: [email protected] Received 5/APR/2012 Originally published in English

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